Stability Analysis of Equilibrium States of an SEIR Tuberculosis Model

Abstract

We extend the tuberculosis model proposed by Blower etal. [1] by incorporating factors such as rates of detection and treatment of active tuberculosis (TB), proportions of recruited individuals due to immigration, rate at which susceptible individuals become infectious and the recovered class. We prove that the solution to the model is positive and bounded. We examine the stability and equilibrium states of the extended model with respect to the basic reproduction number R₀. We show that the disease-free equilibrium (DFE) is globally asymptotically stable if R₀ ≤ 1 and that there exists at least one endemic equilibrium which is globally asymptotically stable if R₀ > 1. Finally, based on our results, we discuss optimum treatment strategies for tuberculosis epidemics.

Keywords: Tuberculosis; Mathematical model; Global stability; Equilibrium; Epidemics; Basic reproduction number

Journal of the Nigerian Association of Mathematical Physics, Volume 20 (March, 2012), pp 119 – 124